FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 
213 
V being the velocity of the charge and square brackets being employed as thronghout 
to denote the vector product. This expression, first given in its complete form by 
Loeentz and Larmor, is generally regarded as being exact.* 
But there are other forms of the additional assumption which are equally effective 
and in some respects more general than that just given. We may, for instance, assume 
definite expressions for the potential and kinetic energies of electromagnetic origin in 
the system and combine these with the assumption that the dynamical processes 
operative in the field are governed by the same general laws as are the processes in a 
similar mechanical system. This form of the argument proves ultimately to be 
consistent with the first as regards the expression for the effective mechanical force on 
a moving element of charge, but it has the advantage of being expressed in more 
general terms, thus carrying with it the possibility of fitting better with any modifi¬ 
cation that it may subsequently be thought desirable to make in our general 
conception of the theory. It need not be presumed that this form of the argument is 
any less general than the first on account of the fact that it apparently involves more 
than one assumption, for this increase in number is counterbalanced by the fact that 
the dynamical argument ultimately reduces the two fundamental equations to one, 
Faraday s relation being derived simultaneously as a consequence of Ampere’s. 
The general dynamical argument was first formulated by MaxwellI for the case 
of the field surrounding a system of linear currents. His analysis was subsequently 
extended to cover the more general case, firstly by von Helmholtz| and Lorentz,§ 
later by Larmor,|| Macdonald,11 Abraham,** and others. In the later investigations 
the whole subject is regarded from the point of view of the theory of electrons, where¬ 
in every manifestation of the field is regarded as arising in the aggregate disposition 
or motion of electronic charges; even in the former investigations, although they are 
apparently of a more general character, certain assumptions are involved which render 
their analyses theoretically effective only under the same restricted circumstances. 
In each of these cases it is assumed, practically speaking, that the potential energy of 
the electromagnetic field is represented by the expression 
and the kinetic energy by 
* Relatively modifications are not here under consideration, 
t ‘ Treatise,’ vol. 2, Ch. VI., VII. 
I ‘ Ann. Rhys. Chem.,’ vol. 47 (1892), p. 1. 
§ ‘ La Theorie Electromagnfitique de Maxwell’ (Leiden, 1892), §§ 55-61. 
II ‘ .^ther and Matter,’ § 50. 
H ‘ Electric Waves,’ Appendix I. 
** ‘ Ann. Rhys.,’ vol. 10 (1903), p. 105. 
